Math 106: Review for Exam I
INTEGRATION TIPS
•Substitution: usually let w= an inside function, especially if w0is also present in the integrand
•Parts: Zudv =uv −Zvdu or Zuv0dx =uv −Zu0vdx
How to choose which part is u? Let ube the part that is higher up in the LIATE mnemonic below.
(The mnemonics ILATE and LIPET will work equally well if you have learned one of those instead;
in the latter Ais replaced by P, which stands for “polynomial”.)
Logarithms (such as ln x)
Inverse trig (such as arctan x, arcsin x)
Algebraic (such as x, x2,x
3+4)
Trig (such as sin x, cos 2x)
Exponentials (such as ex,e
3x)
•Rational Functions (one polynomial divided by another): if the degree of the numerator is greater than
or equal to the degree of the numerator, do long division then integrate the result.
Partial Fractions: here’s an illustrative example of the setup.
3x2+11
(x+ 1)(x−3)2(x2+5) =A
x+1+B
x−3+C
(x−3)2+Dx +E
x2+5
Each linear term in the denominator on the left gets a constant above it on the right; the squared
linear factor (x−3) on the left appears twice on the right, once to the second power. Each irreducible
quadratic term on the left gets a linear term (Dx +Ehere) above it on the right.
1. Find the following.
(a) Z4
1
e√x
√xdx
(b) Zx3ln xdx
(c) Z3x2+2x−13
(x−3)(x2+1)dx
